Why does Carmichael's function mean what it does?

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Sam Anderson

Carmichael's function λ(n) gives smallest number "m" such that if x≡y mod m ⇒ a^x ≡ a^y mod n, but WHY?
How did Carmichael figure this out?

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mfb

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That is a definition, there is nothing to figure out about definitions.
You can figure out how to calculate values of this function.